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Weak Maass-Poincaré series and weight 3/2 mock modular forms. (English) Zbl 1287.11052
J. Number Theory 133, No. 8, 2567-2587 (2013); corrigendum ibid. 159, 434-435 (2016).
Summary: The primary goal of this paper is to construct the basis of the space of weight \(3/2\) mock modular forms, which is an extension of the Borcherds-Zagier basis of weight \(3/2\) weakly holomorphic modular forms. The shadows of the members of this basis form the Borcherds-Zagier basis of the space of weight \(1/2\) weakly holomorphic modular forms. In the course of the construction, we provide a full computation of the Fourier coefficients for the weak Maass-Poincaré series in most general form for the purpose of future reference.

MSC:
11F03 Modular and automorphic functions
11F12 Automorphic forms, one variable
11F30 Fourier coefficients of automorphic forms
11F37 Forms of half-integer weight; nonholomorphic modular forms
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References:
[1] Abramowitz, M.; Stegun, I., Pocketbook of mathematical functions, (1984), Verlag Harri Deutsch
[2] Bringmann, K.; Ono, K., Arithmetic properties of coefficients of half-integral weight Maass-Poincaré series, Math. Ann., 337, 591-612, (2007) · Zbl 1154.11015
[3] Bringmann, K.; Ono, K., Lifting elliptic cusp forms to Maass forms with an application to partitions, Proc. Natl. Acad. Sci. USA, 104, 3725-3731, (2007) · Zbl 1191.11013
[4] Bringmann, K.; Kane, B.; Rhoades, R., Duality and differential operators for harmonic Maass forms, (From Fourier Analysis and Number Theory to Radon Transforms and Geometry, in Memory of Leon Ehrenpreis, Dev. Math., vol. 28, (2013), Springer), 85-166 · Zbl 1282.11043
[5] Bruggeman, R., Harmonic lifts of modular forms · Zbl 1311.11033
[6] Bruinier, J. H., Borcherds products on \(O(2, \ell)\) and Chern classes of Heegner divisors, Springer Lect. Notes, vol. 1780, (2002), Springer-Verlag Berlin · Zbl 1004.11021
[7] Bruinier, J. H.; Funke, J., On two geometric theta lifts, Duke Math. J., 125, 1, 45-90, (2004) · Zbl 1088.11030
[8] Bruinier, J. H.; Jenkins, P.; Ono, K., Hilbert class polynomials and traces of singular moduli, Math. Ann., 334, 373-393, (2006) · Zbl 1091.11011
[9] Duke, W.; Imamoḡlu, Ö.; Tóth, Á., Cycle integrals of the j-function and mock modular forms, Ann. Math., 173, 2, 947-981, (2011) · Zbl 1270.11044
[10] W. Duke, Ö. Imamoḡlu, Á. Tóth, Weight two weakly harmonic Maass forms, preprint. · Zbl 1213.11141
[11] Fay, J., Fourier coefficients of the resolvent for a Fuchsian group, J. Reine Angew. Math., 294, 143-203, (1977) · Zbl 0352.30012
[12] Hejhal, D. A., The Selberg trace formula for \(\mathit{PSL}_2(\mathbb{R})\), Springer Lect. Notes, vol. 1001, (1983), Springer-Verlag Berlin
[13] Jeon, D.; Kang, S.-Y.; Kim, C. H., Cycle integrals of a sesqui-harmonic Maass form of weight zero, preprint · Zbl 1322.11036
[14] Kang, S.-Y.; Kim, C. H., Arithmetic properties of traces of singular moduli on congruence subgroups, Int. J. Number Theory, 6, 8, 1755-1768, (2010) · Zbl 1221.11101
[15] Kohnen, W., Fourier coefficients of modular forms of half-integral weight, Math. Ann., 271, 237-268, (1985) · Zbl 0542.10018
[16] Miller, A.; Pixton, A., Arithmetic traces of non-holomorphic modular invariants, Int. J. Number Theory, 6, 1, 69-87, (2010) · Zbl 1245.11061
[17] Neunhöffer, H., Über die analytische fortsetzung von poincarreihen, S.-B. Heidelberger Akad. Wiss. Math.-Natur. Kl., 33-90, (1973), (in German)
[18] Niebur, D., A class of nonanalytic automorphic functions, Nagoya Math. J., 52, 133-145, (1973) · Zbl 0288.10010
[19] Zagier, D., Traces of singular moduli, (Motives, Polylogarithms and Hodge Theory, Part I, Irvine, CA, 1998, Int. Press Lect. Ser. 3, vol. I, (2002), International Press Somerville, MA), 211-244 · Zbl 1048.11035
[20] Zagier, D., Ramanujanʼs mock theta functions and their applications, in: Séminaire Bourbaki 60ème année (986), 2006-2007
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